System and computer-implemented method for evaluating integrals using a quasi-monte carlo methodology in which sample points represent dependent samples generated using a low-discrepancy sequence

ABSTRACT

A computer graphics system generates a pixel value for a pixel in an image, the pixel value being representative of a point in a scene as recorded on an image plane of a simulated camera, the computer graphics system comprising a sample point generator and a function evaluator. The sample point generator is configured to generate a set of sample points, at least one sample point being generated using at least one dependent sample, the at least one dependent sample comprising at least one element of a low-discrepancy sequence offset by at least one element of another low-discrepancy sequence. The function evaluator is configured to generate at least one value representing an evaluation of a selected function at one of the sample points generated by the sample point generator, the value generated by the function evaluator corresponding to the pixel value.

INCORPORATION BY REFERENCE

U.S. patent application Ser. No. 08/880,418, filed Jun. 23, 1997, in the names of Martin Grabenstein, et al., entitled “System And Method For Generating Pixel Values For Pixels In An Image Using Strictly Deterministic Methodologies For Generating Sample Points,” (hereinafter referred to as the Grabenstein application) assigned to the assignee of this application, incorporated by reference.

FIELD OF THE INVENTION

The invention relates generally to the field of systems and computer-implemented methods for evaluating integrals, and more particularly to such systems and computer-implemented methods that evaluate integrals using quasi-Monte Carlo methodologies. The invention particularly provides a methodology that makes use of trajectory splitting by dependent sampling. Systems and computer-implemented methods according to the invention find utility in a number of applications, including but not limited to computer graphics.

BACKGROUND OF THE INVENTION

In computer graphics, a computer is used to generate digital data that represents the projection of surfaces of objects in, for example, a three-dimensional scene, illuminated by one or more light sources, onto a two-dimensional image plane, to simulate the recording of the scene by, for example, a camera. The camera may include a lens for projecting the image of the scene onto the image plane, or it may comprise a pinhole camera in which case no lens is used. The two-dimensional image is in the form of an array of picture elements (which are variable termed “pixels” or “pels”), and the digital data generated for each pixel represents the color and luminance of the scene as projected onto the image plane at the point of the respective pixel in the image plane. The surfaces of the objects may have any of a number of characteristics, including shape, color, specularity, texture, and so forth, which are preferably rendered in the image as closely as possible, to provide a realistic-looking image.

Generally, the contributions of the light reflected from the various points in the scene to the pixel value representing the color and intensity of a particular pixel are expressed in the form of the one or more integrals of relatively complicated functions. Since the integrals used in computer graphics generally will not have a closed-form solution, numerical methods must be used to evaluate them and thereby generate the pixel value. Typically, a conventional “Monte Carlo” method has been used in computer graphics to numerically evaluate the integrals. Generally, in the Monte Carlo method, to evaluate an integral $\begin{matrix} {\left\langle f \right\rangle = {\int_{{\lbrack{0,1})}^{s}}{{f(x)}{\mathbb{d}x}}}} & (1) \end{matrix}$ where f(x) is a real function on the “s”-dimensional unit cube [0,1)^(s) (that is, an s-dimensional cube each of whose dimension includes “zero,” and excludes “one”), first a number “N” statistically-independent randomly-positioned points xi, i=1, . . . , N, are generated over the integration domain. The random points xi are used as sample points for which sample values f(xi) are generated for the function f(x), and an estimate {overscore (ƒ)} for the integral is generated as $\begin{matrix} {{\left\langle f \right\rangle \approx \overset{\_}{f}} = {\frac{1}{N}{\sum\limits_{i = 1}^{N}{f\left( x_{i} \right)}}}} & (2) \end{matrix}$ As the number of random points used in generating the sample points f(xi) increases, the value of the estimate {overscore (ƒ)} will converge toward the actual value of the integral

ƒ

. Generally, the distribution of estimate values that will be generated for various values of “N,” that is, for various numbers of sample points, of being normal distributed around the actual value with a standard deviation σ which can be estimated by $\begin{matrix} {\sigma = \sqrt{\frac{1}{N - 1}\left( {{\overset{\_}{f}}^{2} - {\overset{\_}{f}}^{2}} \right)}} & (3) \end{matrix}$ if the points x_(i) used to generate the sample values f(x_(i)) are statistically independent, that is, if the points x_(i) are truly positioned at random in the integration domain.

Generally, it has been believed that random methodologies like the Monte Carlo method are necessary to ensure that undesirable artifacts, such as Moiré patterns and aliasing and the like, which are not in the scene, will not be generated in the generated image. However, several problems arise from use of the Monte Carlo method in computer graphics. First, since the sample points x_(i) used in the Monte Carlo method are randomly distributed, they may clump in various regions over the domain over which the integral is to be evaluated. Accordingly, depending on the set of points that are generated, in the Monte Carlo method for significant portions of the domain there may be no sample points x_(i) for which sample values f(x_(i)) are generated. In that case, the error can become quite large. In the context of generating a pixel value in computer graphics, the pixel value that is actually generated using the Monte Carlo method may not reflect some elements which might otherwise be reflected if the sample points x_(i) were guaranteed to be more evenly distributed over the domain. This problem can be alleviated somewhat by dividing the domain into a plurality of sub-domains, but it is generally difficult to determine a priori the number of sub-domains into which the domain should be divided, and, in addition, in a multi-dimensional integration region, which would actually be used in computer graphics rendering operations, the partitioning of the integration domain into sub-domains, which are preferably of equal size, can be quite complicated.

In addition, since the method makes use of random numbers, the error |{overscore (ƒ)}−

ƒ

| (where |x| represents the absolute value of the value “x”) between the estimate value {overscore (ƒ)} and actual value

ƒ

is probabilistic, and, since the error values for various large values of “N” are close to normal distribution around the actual value

ƒ

, only sixty-eight percent of the estimate values {overscore (ƒ)} that might be generated are guaranteed to lie within one standard deviation of the actual value

ƒ

.

Furthermore, as is clear from equation (3), the standard deviation σ decreases with increasing numbers “N” of sample points, proportional to the reciprocal of square root of “N” (that is, 1/{square root}{square root over (N)}). Thus, if it is desired to reduce the statistical error by a factor of two, it will be necessary to increase the number of sample points N by a factor of four, which, in turn, increases the computational load that is required to generate the pixel values, for each of the numerous pixels in the image.

Additionally, since the Monte Carlo method requires random numbers to define the coordinates of respective sample points x_(i) in the integration domain, an efficient mechanism for generating random numbers is needed. Generally, digital computers are provided with so-called “random number” generators, which are computer programs which can be processed to generate a set of numbers that are approximately random. Since the random number generators use deterministic techniques, the numbers that are generated are not truly random. However, the property that subsequent random numbers from a random number generator are statistically independent should be maintained by deterministic implementations of pseudo-random numbers on a computer.

The Grabenstein application describes a computer graphics system and method for generating pixel values for pixels in an image using a strictly deterministic methodology for generating sample points, which avoids the above-described problems with the Monte Carlo method. The strictly deterministic methodology described in the Grabenstein application provides a low-discrepancy sample point sequence which ensures, a priori, that the sample points are generally more evenly distributed throughout the region over which the respective integrals are being evaluated. In one embodiment, the sample points that are used are based on a so-called Halton sequence. See, for example, J. H. Halton, Numerische Mathematik, Vol. 2, pp. 84-90 (1960) and W. H. Press, et al., Numerical Recipes in Fortran (2d Edition) page 300 (Cambridge University Press, 1992). In a Halton sequence generated for number base “b,” where base “b” is a selected prime number, the “k-th” value of the sequence, represented by H_(b) ^(k) is generated by use of a “radical inverse” function radical inverse function Φ_(b) that is generally defined as $\begin{matrix} {{{\Phi_{b}\text{:}N_{0}}->I}{i = \left. {\sum\limits_{j = 0}^{\infty}{{a_{j}(i)}b^{j}}}\mapsto{\sum\limits_{j = 0}^{\infty}{{a_{j}(i)}b^{{- j} - 1}}} \right.}} & (4) \end{matrix}$ where (a_(j))_(j=0) ^(∞) is the representation of “i” in integerbase “b.” Generally, a radical inverse of a value “k” is generated by

(1) writing the value “k” as a numerical representation of the value in the selected base “b,” thereby to provide a representation for the value as D_(M)D_(M−1) . . . D₂D₁, where D_(m) (m=1, 2, . . . , M) are the digits of the representation,

(2) putting a radix point (corresponding to a decimal point for numbers written in base ten) at the least significant end of the representation D_(M)D_(M−1) . . . D₂D₁ written in step (1) above, and

(3) reflecting the digits around the radix point to provide 0.D₁D₂ . . . D_(M−1)D_(M), which corresponds to H_(b) ^(k).

It will be appreciated that, regardless of the base “b” selected for the representation, for any series of values, one, two, . . . “k,” written in base “b,” the least significant digits of the representation will change at a faster rate than the most significant digits. As a result, in the Halton sequence H_(b) ¹, H_(b) ², . . . H_(b) ^(k), the most significant digits will change at the faster rate, so that the early values in the sequence will be generally widely distributed over the interval from zero to one, and later values in the sequence will fill in interstices among the earlier values in the sequence. Unlike the random or pseudo-random numbers used in the Monte Carlo method as described above, the values of the Halton sequence are not statistically independent; on the contrary, the values of the Halton sequence are strictly deterministic, “maximally avoiding” each other over the interval, and so they will not clump, whereas the random or pseudo-random numbers used in the Monte Carlo method may clump.

It will be appreciated that the Halton sequence as described above provides a sequence of values over the interval from zero to one, inclusive along a single dimension. A multi-dimensional Halton sequence can be generated in a similar manner, but using a different base for each dimension, where the bases are relatively prime.

A generalized Halton sequence, of which the Halton sequence described above is a special case, is generated as follows. For each starting point along the numerical interval from zero to one, inclusive, a different Halton sequence is generated. Defining the pseudo-sum x⊕_(p)y for any x and y over the interval from zero to one, inclusive, for any integer “p” having a value greater than two, the pseudo-sum is formed by adding the digits representing “x” and “y” in reverse order, from the most-significant digit to the least-significant digit, and for each addition also adding in the carry generated from the sum of next more significant digits. Thus, if “x” in base “b” is represented by 0.X₁X₂ . . . X_(M−1)X_(M), where each “X_(m)” is a digit in base “b,” and if “y” in base “b” is represented by 0.Y₁Y₂ . . . Y_(N−1)Y_(N), where each “Y_(n)” is a digit in base “b” (and where “M,” the number of digits in the representation of “x” in base “b”, and “N,” the number of digits in the representation of “y” in base “b”, may differ), then the pseudo-sum “z” is represented by 0.Z₁Z₂ . . . Z_(L−1)Z_(L), where each “Z₁” is a digit in base “b” given by Z₁=(X₁+Y₁+C₁) mod b, where “mod” represents the modulo function, and $C_{l} = \left\{ \begin{matrix} {{{1\quad{for}\quad X_{l - 1}} + Y_{l - 1} + Z_{l - 1}} \geq b} \\ {0\quad{otherwise}} \end{matrix} \right.$ is a carry value from the “1-1st” digit position, with C₁ being set to zero.

Using the pseudo-sum function as described above, the generalized Halton sequence that is used in the system described in the Grabenstein application is generated as follows. If “b” is an integer, and x₀ is an arbitrary value on the interval from zero to one, inclusive, then the “p”-adic von Neumann-Kakutani transformation T_(b)(x) is given by $\begin{matrix} {{{T_{p}(x)}:={x \oplus_{p}\frac{1}{b}}},} & (5) \end{matrix}$ and the generalized Halton sequence x₀, x₁, x_(2, . . .) is defined recursively as x _(n+1) =T _(b)(x _(n))  (6) From equations (5) and (6), it is clear that, for any value for “b,” the generalized Halton sequence can provide that a different sequence will be generated for each starting value of “x,” that is, for each x₀. It will be appreciated that the Halton sequence H_(b) ^(k) as described above is a special case of the generalized Halton sequence (equations (5) and (6)) for x₀=0.

The use of a strictly deterministic low-discrepancy sequence such as the Halton sequence or the generalized Halton sequence can provide a number of advantages over the random or pseudo-random numbers that are used in connection with the Monte Carlo technique. Unlike the random numbers used in connection with the Monte Carlo technique, the low discrepancy sequences ensure that the sample points are more evenly distributed over a respective region or time interval, thereby reducing error in the image which can result from clumping of such sample points which can occur in the Monte Carlo technique. That can facilitate the generation of images of improved quality when using the same number of sample points at the same computational cost as in the Monte Carlo technique.

SUMMARY OF THE INVENTION

The invention provides a new and improved system and computer-implemented method for evaluating integrals using a quasi-Monte Carlo methodology that makes use of trajectory splitting by dependent sampling.

In brief summary, the invention provides a computer graphics system for generating a pixel value for a pixel in an image, the pixel value being representative of a point in a scene as recorded on an image plane of a simulated camera, the computer graphics system comprising a sample point generator and a function evaluator. The sample point generator is configured to generate a set of sample points, at least one sample point being generated using at least one dependent sample, the at least one dependent sample comprising at least one element of a low-discrepancy sequence offset by at least one element of another low-discrepancy sequence. The function evaluator is configured to generate at least one value representing an evaluation of a selected function at one of the sample points generated by the sample point generator, the value generated by the function evaluator corresponding to the pixel value.

BRIEF DESCRIPTION OF THE DRAWINGS

This invention is pointed out with particularity in the appended claims. The above and further advantages of this invention may be better understood by referring to the following description taken in conjunction with the accompanying drawings, in which:

FIG. 1 depicts an illustrative computer graphics system that evaluates integrals using a quasi-Monte Carlo methodology that makes use of trajectory splitting by dependent sampling.

DETAILED DESCRIPTION OF AN ILLUSTRATIVE EMBODIMENT

The invention provides an computer graphic system and method for generating pixel values for pixels in an image of a scene, which makes use of a strictly-deterministic quasi-Monte Carlo methodology that makes use of trajectory splitting by dependent sampling for generating sample points for use in generating sample values for evaluating the integral or integrals whose function(s) represent the contributions of the light reflected from the various points in the scene to the respective pixel value, rather than the random or pseudo-random Monte Carlo methodology which has been used in the past. The strictly-deterministic methodology ensures a priori that the sample points will be generally more evenly distributed over the interval or region over which the integral(s) is (are) to be evaluated in a low-discrepancy manner.

FIG. 1 attached hereto depicts an illustrative computer system 10 that makes use of such a strictly deterministic methodology. With reference to FIG. 1, the computer system 10 in one embodiment includes a processor module 11 and operator interface elements comprising operator input components such as a keyboard 12A and/or a mouse 12B (generally identified as operator input element(s) 12) and an operator output element such as a video display device 13. The illustrative computer system 10 is of the conventional stored-program computer architecture. The processor module 11 includes, for example, one or more processor, memory and mass storage devices, such as disk and/or tape storage elements (not separately shown), which perform processing and storage operations in connection with digital data provided thereto. The operator input element(s) 12 are provided to permit an operator to input information for processing. The video display device 13 is provided to display output information generated by the processor module 11 on a screen 14 to the operator, including data that the operator may input for processing, information that the operator may input to control processing, as well as information generated during processing. The processor module 11 generates information for display by the video display device 13 using a so-called “graphical user interface” (“GUI”), in which information for various applications programs is displayed using various “windows.” Although the computer system 10 is shown as comprising particular components, such as the keyboard 12A and mouse 12B for receiving input information from an operator, and a video display device 13 for displaying output information to the operator, it will be appreciated that the computer system 10 may include a variety of components in addition to or instead of those depicted in FIG. 1.

In addition, the processor module 11 includes one or more network ports, generally identified by reference numeral 14, which are connected to communication links which connect the computer system 10 in a computer network. The network ports enable the computer system 10 to transmit information to, and receive information from, other computer systems and other devices in the network. In a typical network organized according to, for example, the client-server paradigm, certain computer systems in the network are designated as servers, which store data and programs (generally, “information”) for processing by the other, client computer systems, thereby to enable the client computer systems to conveniently share the information. A client computer system which needs access to information maintained by a particular server will enable the server to download the information to it over the network. After processing the data, the client computer system may also return the processed data to the server for storage. In addition to computer systems (including the above-described servers and clients), a network may also include, for example, printers and facsimile devices, digital audio or video storage and distribution devices, and the like, which may be shared among the various computer systems connected in the network. The communication links interconnecting the computer systems in the network may, as is conventional, comprise any convenient information-carrying medium, including wires, optical fibers or other media for carrying signals among the computer systems. Computer systems transfer information over the network by means of messages transferred over the communication links, with each message including information and an identifier identifying the device to receive the message.

It will be helpful to initially provide some background on operations performed by the computer graphics system in generating an image. Generally, the computer graphic system generates an image that attempts to simulate an image of a scene that would be generated by a camera. The camera includes a shutter that will be open for a predetermined time T starting at a time t₀ to allow light from the scene to be directed to an image plane. The camera may also include a lens or lens model (generally, “lens”) that serves to focus light from the scene onto the image plane. The average radiance flux L_(m,n) through a pixel at position (m,n) on an image plane P, which represents the plane of the camera's recording medium, is determined by $\begin{matrix} {L_{m,n} = {\frac{1}{{A_{P}} \cdot T \cdot {A_{L}}}{\int_{A_{P}}{\int_{t_{0}}^{t_{0} + T}{\int_{A_{L}}{{L\left( {{h\left( {x,t,y} \right)},\quad{- {\omega\left( {x,t,y} \right)}}} \right)}{f_{m,n}\left( {x,y,t} \right)}{\mathbb{d}y}{\mathbb{d}t}{\mathbb{d}x}}}}}}} & (7) \end{matrix}$ where “A_(p)” refers to the area of the pixel, A_(L) refers to the area of the portion of the lens through which rays of light pass from the scene to the pixel, and f_(m,n) represents a filtering kernel associated with the pixel. An examination of the integral in equation (7) will reveal that, for the variables of integration, “x,” “y” and “t,” the variable “y” refers to integration over the lens area (A_(L)), the variable “t” refers to integration over time (the time interval from t₀ to t₀+T) and the variable “x” refers to integration over the pixel area (A_(p)).

The value of the integral in equation (7) is approximated in accordance with a quasi-Monte Carlo methodology by identifying N_(p) sample points x_(i) in the pixel area, and, for each sample point, shooting N_(T) rays at times t_(i,j) in the time interval to t₀ to t₀+T through the focus into the scene, with each ray spanning N_(L) sample points y_(i,j,k) on the lens area A_(L). The manner in which subpixel jitter positions x_(i), points in time t_(i,j) and positions on the lens y_(i,j,k) are determined will be described below. These three parameters determine the primary ray hitting the scene geometry in h(x_(i),t_(i,j),y_(i,j,k)) with the ray direction ω(x_(i),t_(i,j),y_(i,j,k)). In this manner, the value of the integral in equation (7) can be approximated as $\begin{matrix} {{L_{m,n} \approx {\frac{1}{N}\underset{i = 0}{\overset{N_{P} - 1}{\sum\quad}}\frac{1}{N_{T}}\underset{j = 0}{\overset{N_{T} - 1}{\sum\quad}}\frac{1}{N_{L}}{\sum\limits_{k = 0}^{N_{L} - 1}{{L\left( {{h\left( {x_{i},t_{i,j},y_{i,j,k}} \right)},\quad{- {\omega\left( {x_{i},{t_{i,j}y_{i,j,k}}} \right)}}} \right)}{f_{m,n}\left( {x_{i},t_{i,j},y_{i,j,k}} \right)}}}}},} & (8) \end{matrix}$ where “N” is the total number of rays directed at the pixel.

It will be appreciated that rays directed from the scene toward the image plane can comprise rays directly from one or more light sources in the scene, as well as rays reflected off surfaces of objects in the scene. In addition, it will be appreciated that a ray that is reflected off a surface may have been directed to the surface directly from a light source, or a ray that was reflected off another surface. For a surface that reflects light rays, a reflection operator T_(fr) is defined that includes a diffuse portion T_(fd), a glossy portion T_(fg) and a specular portion T_(fs), or T _(f) _(r) =T _(f) _(d) +T _(f) _(g) +T _(f) _(s)   (9). In that case, the Fredholm integral equation L=L_(e)+T_(fr)L governing light transport can be represented as L=L _(e) +T _(f) _(r) _(−f) _(s) L _(e) +T _(f) _(s) (L−L _(e))+T _(f) _(s) L+T _(f) _(d) T _(f) _(g) _(f+) _(s) L+T _(f) _(d) T _(f) _(d) L  (10), where transparency has been ignored for the sake of simplicity; transparency is treated in an analogous manner. The individual terms in equation (10) are

(i) L_(e) represents flux due to a light source;

(ii) T_(f) _(r) _(−f) _(s) L_(e) (where T_(f) _(r) _(−f) _(s) T_(f) _(r) =−T_(f) _(s) ) represents direct illumination, that is, flux reflected off a surface that was provided thereto directly by a light source; the specular component, associated with the specular portion T_(fs) of the reflection operator, will be treated separately since it is modeled using a δ-distribution;

(iii) T_(f) _(g) (L−L_(e)) represents glossy illumination, which is handled by recursive distribution ray tracing, where, in the recursion, the source illumination has already been accounted for by the direct illumination (item (ii) above);

(iv) T_(f) _(s) L represents a specular component, which is handled by recursively using “L” for the reflected ray;

(v) T_(f) _(d) T_(f) _(g) _(+f) _(s) L (where T_(f) _(g) _(+f) _(s) =T_(f) _(g) +T_(f) _(s) ) represents a caustic component, which is a ray that has been reflected off a glossy or specular surface (reference the T_(f) _(g) _(+f) _(s) operator) before hitting a diffuse surface (reference the T_(f) _(d) operator); this contribution can be approximated by a high resolution caustic photon map; and

(vi) T_(f) _(d) T_(f) _(d) L represents ambient light, which is very smooth and is therefore approximated using a low resolution global photon map.

As noted above, the value of the integral (equation (7)) is approximated by solving equation (8) making use of sample points x_(i), t_(i,j) and y_(i,j,k), where “x_(i)” refers to sample points within area A_(L) of the respective pixel at location (m,n) in the image plane, “t_(i,j)” refers to sample points within the time interval t₀ to t₀+T during which the shutter is open, and “y_(i,j,k)” refers to sample points on the lens A_(L). In accordance with one aspect of the invention, the sample points x_(i) comprise two-dimensional Hammersley points, which are defined as $\left( {\frac{i}{N},{\Phi_{2}(i)}} \right),$ where 0≦i<N, and Φ₂(i) refers to the radical inverse of “i” in base “two.” Generally, the “s” dimensional Hammersley point set is defined $\begin{matrix} {{{U_{N,s}^{Hammersley}\text{:}\left\{ {0,\cdots,{N - 1}} \right\}}->I^{s}}{{\left. i\mapsto x_{i} \right.:=\left( {\frac{i}{N},{\Phi_{b_{1}}(i)},\cdots,{\Phi_{b_{s - 1}}(i)}} \right)},}} & (11) \end{matrix}$ where I^(s) is the s-dimensional unit cube [0,1)^(s) (that is, an s-dimensional cube each of whose dimension includes “zero,” and excludes “one”), the number of points “N” in the set is fixed and b₁, . . . ,b_(s−1) are bases. The bases do not need to be prime numbers, but they are preferably relatively prime to provide a uniform distribution. The radical inverse function Φ_(b), in turn, is generally defined as $\begin{matrix} {{{\Phi_{b}:N_{0}}->I}{i = \left. {\sum\limits_{j = 0}^{\infty}{{a_{j}(i)}b^{j}}}\mapsto{\sum\limits_{j = 0}^{\infty}{{a_{j}(i)}b^{{- j} - 1}}} \right.}} & (12) \end{matrix}$ where (a_(j))_(j=0) ^(∞) is the representation of “i” in integer base “b.” At N=(2^(n))², the two-dimensional Hammersley points are a (0, 2n, 2)-net in base “two,” which are stratified on a 2^(n) by 2^(n) grid and a Latin hypercube sample at the same time. Considering the grid as subpixels, the complete subpixel grid underlying the image plane can be filled by simply abutting copies of the grid to each other.

Given integer subpixel coordinates (s_(x),s_(y)) the instance “i” and coordinates (x,y) for the sample point x_(i) in the image plane can be determined as follows. Preliminarily, examining $\left( {\frac{i}{N},{\Phi_{2}(i)}} \right)_{i = 0}^{N - 1},$ one observes that

(a) each line in the stratified pattern is a shifted copy of another, and

(b) the pattern is symmetric to the line y=x, that is, each column is a shifted copy of another column.

Accordingly, given the integer permutation σ(k):=2^(n)Φ₂(k) for 0≦k<2^(n), subpixel coordinates (s_(x),s_(y)) are mapped onto strata coordinates (j,k):=(s_(x) mod 2^(n), s_(y) mod 2^(n)), an instance number “i” is computed as i=j2^(n)+σ(k)  (13) and the positions of the jittered subpixel sample points are determined according to $\begin{matrix} \begin{matrix} {x_{i} = \left( {{s_{x} + {\Phi_{2}(k)}},{s_{y} + {\Phi_{2}(j)}}} \right)} \\ {= {\left( {{s_{x} + \frac{\sigma(k)}{2^{n}}},{s_{y} + \frac{\sigma(j)}{2^{n}}}} \right).}} \end{matrix} & (14) \end{matrix}$ An efficient algorithm for generating the positions of the jittered subpixel sample points x_(i) will be provided below in connection with Code Segment 1. A pattern of sample points whose positions are determined as described above in connection with equations (13) and (14) has an advantage of having much reduced discrepancy over a pattern determined using a Halton sequence or windowed Halton sequence, as described in the aforementioned Grabenstein application, and therefore the approximation described above in connection with equation (8) gives in general a better estimation to the value of the integral described above in connection with equation (7). In addition, if “N” is sufficiently large, sample points in adjacent pixels will have different patterns, reducing the likelihood that undesirable artifacts will be generated in the image.

A ray tree is a collection of paths of light rays that are traced from a point on the simulated camera's image plane into the scene. The computer graphics system 10 generates a ray tree by recursively following transmission, subsequent reflection and shadow rays using trajectory splitting. In accordance with another aspect of the invention, a path is determined by the components of one vector of a global generalized scrambled Hammersley point set. Generally, a scrambled Hammersley point set reduces or eliminates a problem that can arise in connection with higher-dimensioned low-discrepancy sequences since the radical inverse function Φ_(b) typically has subsequences of b−1 equidistant values spaced by 1/b. Although these correlation patterns are merely noticeable in the full s-dimensional space, they are undesirable since they are prone to aliasing. The computer graphics system 10 attenuates this effect by scrambling, which corresponds to application of a permutation to the digits of the b-ary representation used in the radical inversion. For the symmetric permutation σ from the symmetric group S_(b) over integers 0, . . . , b−1, the scrambled radical inverse is defined as $\begin{matrix} {{{\Phi_{b}:{N_{0} \times S_{b}}}->\left. {I\left( {i,\sigma} \right)}\mapsto{\sum\limits_{j = 0}^{\infty}{{\sigma\left( {a_{j}(i)} \right)}\left. b^{{- j} - 1}\Longleftrightarrow i \right.}} \right.} = {\sum\limits_{j = 0}^{\infty}{{a_{j}(i)}{b^{j}.}}}} & (15) \end{matrix}$ If the symmetric permutation “σ” is the identity, the scrambled radical inverse corresponds to the unscrambled radical inverse. In one embodiment, the computer graphics system generates the symmetric permutation a recursively as follows. Starting from the permutation σ₂=(0,1) for base b=2, the sequence of permutations is defined as follows:

(i) if the base “b” is even, the permutation σ_(b) is generated by first taking the values of 2σ_(b/2) and appending the values of ${{2\sigma_{\frac{b}{2}}} + 1},$ and

(ii) if the base “b” is odd, the permutation σ_(b) is generated by taking the values of σ_(b−1), incrementing each value that is greater than or equal to $\frac{b - 1}{2}$ by one, and inserting the value $\frac{b - 1}{2}$ in the middle. This recursive procedure results in

-   -   σ₂=(0,1)     -   σ₃=(0,1,2)     -   σ₄=(0,2,1,3)     -   σ₅=(0,3,2,1,4)     -   σ₆=(0,2,4,1,3,5)     -   σ₇=(0,2,5,3,1,4,6)     -   σ₈=(0,4,2,6,1,5,3,7) . . .

The computer graphics system 10 can generate a generalized low-discrepancy point set as follows. It is often possible to obtain a low-discrepancy sequence by taking any rational s-dimensional point “x” as a starting point and determine a successor by applying the corresponding incremental radical inverse function to the components of “x.” The result is referred to as the generalized low-discrepancy point set. This can be applied to both the Halton sequence and the Hammersley sequence. In the case of the generalized Halton sequence, this can be formalized as x _(i)(Φ_(b) ₁ (i+i ₁),Φ_(b) ₂ (i+i ₂), . . . , Φ(i+i _(s)))  (16), where the integer vector (i₁, i₂, . . . , i_(s)) represents the offsets per component and is fixed in advance for a generalized sequence. The integer vector can be determined by applying the inverse of the radical inversion to the starting point “x.” A generalized Hammersley sequence can be generated in an analogous manner.

Returning to trajectory splitting, generally trajectory splitting is the evaluation of a local integral, which is of small dimension and which makes the actual integrand smoother, which improves overall convergence. Applying replication, positions of low-discrepancy sample points are determined that can be used in evaluating the local integral. The low-discrepancy sample points are shifted by the corresponding elements of the global scrambled Hammersley point set. Since trajectory splitting can occur multiple times on the same level in the ray tree, branches of the ray tree are decorrelated in order to avoid artifacts, the aecorrelation being accomplished by generalizing the global scrambled Hammersley point set.

An efficient algorithm for generating a ray tree will be provided below in connection with Code Segment 2. Generally, in that algorithm, the instance number “i” of the low-discrepancy vector, as determined above in connection with equation (13), and the number “d” of used components, which corresponds to the current integral dimension, are added to the data structure that is maintained for the respective ray in the ray tree. The ray tree of a subpixel sample is completely specified by the instance number “i.” After the dimension has been set to “two,” which determines the component of the global Hammersley point set that is to be used next, the primary ray is cast into the scene to span its ray tree. In determining the deterministic splitting by the components of low discrepancy sample points, the computer graphics system 10 initially allocates the required number of dimensions “Δd.” For example, in simulating glossy scattering, the required number of dimensions will correspond to “two.” Thereafter, the computer graphics system 10 generates scattering directions from the offset given by the scrambled radical inverses Φ_(b) _(d) (i,σ_(b) _(d) ), . . . , Φ_(b) _(d+Δd−1) (i,σ_(b) _(d+Δd−1) ), yielding the instances $\begin{matrix} {{\left( y_{i,j} \right)_{j = 0}^{M - 1} = \left( {{{\Phi_{b_{d}}\left( {i,\sigma_{b_{d}}} \right)} \oplus \frac{j}{M}},\cdots\quad,{{\Phi_{b_{d + {\Delta\quad d} - 1}}\left( {i,\sigma_{b_{d + {\Delta\quad d} - 1}}} \right)} \oplus {\Phi_{b_{d + {\Delta\quad d} - 2}}\left( {j,\sigma_{b_{d + {\Delta\quad d} - 2}}} \right)}}} \right)},} & (17) \end{matrix}$ where “⊕” refers to addition modulo “one.” Each direction of the “M” replicated rays is determined by y_(i,j) and enters the next level of the ray tree with d′:=d+Δd as the new integral dimension in order to use the next elements of the low-discrepancy vector, and i′=i+j in order to decorrelate subsequent trajectories. Using an infinite sequence of low-discrepancy sample points, the replication heuristic is turned into an adaptive consistent sampling arrangement. That is, computer graphics system 10 can fix the sampling rate ΔM, compare current and previous estimates every ΔM samples, and, if the estimates differ by less than a predetermined threshold value, terminate sampling. The computer graphics system 10 can, in turn, determine the threshold value, by importance information, that is, how much the local integral contributes to the global integral.

As noted above, the integral described above in connection with equation (7) is over a finite time period T from t₀ to t₀+T, during which time the shutter of the simulated camera is open. During the time period, if an object in the scene moves, the moving object may preferably be depicted in the image as blurred, with the extent of blurring being a function of the object's motion and the time interval t₀+T. Generally, motion during the time an image is recorded is linearly approximated by motion vectors, in which case the integrand (equation (7)) is relatively smooth over the time the shutter is open and is suited for correlated sampling. For a ray instance “i,” started at the subpixel position x_(i), the offset Φ₃(i) into the time interval is generated and the N_(T)−1 subsequent samples ${\Phi_{3}(i)} + \frac{j}{N_{T}}$ mod 1 are generated for 0<j<N_(T) that is $\begin{matrix} {t_{i,j}:={{t_{0}\left( {{\Phi_{3}(i)} \oplus \frac{j}{N_{T}}} \right)} \cdot {T.}}} & (18) \end{matrix}$ It will be appreciated that the value of N_(T) may be chosen to be “one,” in which case there will be no subsequent samples for ray instance “i.” Determining sample points in this manner fills the sampling space, resulting in a more rapid convergence to the value of the integral (equation (7)). For subsequent trajectory splitting, rays are decorrelated by setting the instance i′=i+j.

In addition to determining the position of the jittered subpixel sample point x_(i), and adjusting the camera and scene according to the sample point t_(i,j) for the time, the computer graphics system also simulates depth of field. In simulating depth of field, the camera to be simulated is assumed to be provided with a lens having known optical characteristics and, using geometrical optics, the subpixel sample point x_(i) is mapped through the lens to yield a mapped point x_(i)′. The lens is sampled by mapping the dependent samples $\begin{matrix} {y_{i,j,k} = \left( {\left( {{\Phi_{5}\left( {{i + j},\sigma_{5}} \right)} \oplus \frac{k}{N_{L}}} \right),\left( {{\Phi_{7}\left( {{i + j},\sigma_{7}} \right)} \oplus {\Phi_{2}(k)}} \right)} \right)} & (19) \end{matrix}$ onto the lens area A_(L) using a suitable one of a plurality of known transformations. As with N_(T), the value of N_(L) may be chosen to be “one.” Thereafter, a ray is shot from the sample point on the lens specified by y_(i,j,k) through the point x_(i)′ into the scene. The offset (Φ₅(i+j,σ₅),Φ₇(i+j,σ₇)) in equation (19) comprises the next components taken from the generalized scrambled Hammersley point set, which, for trajectory splitting, is displaced by the elements $\left( {\frac{k}{N_{L}},{\Phi_{2}(k)}} \right)$ of the two-dimensional Hammersley point set. The instance of the ray originating from sample point y_(i,j,k) is set to “i+j+k” in order to decorrelate further splitting down the ray tree. In equation (19), the scrambled samples (Φ₅(i+j,σ₅),Φ₇(i+j,σ₇)) are used instead of the unscrambled samples of (Φ₅(i+j),Φ₇(i+j)) since in bases “five” and “seven” up to five unscrambled samples will lie on a straight line, which will not be the case for the scrambled samples.

In connection with determination of a value for the direct illumination (T_(f) _(r) _(−f) _(s) L_(e) above), direct illumination is represented as an integral over the surface of the scene ∂V, which integral is decomposed into a sum of integrals, each over one of the “L” single area light sources in the scene. The individual integrals in turn are evaluated by dependent sampling, that is $\begin{matrix} \begin{matrix} {{\left( {T_{f_{r} - f_{s}}L_{e}} \right)\left( {y,z} \right)} = {\int_{\partial V}{{L_{e}\left( {x,y} \right)}{f_{r}\left( {x,y,z} \right)}{G\left( {x,y} \right)}{\mathbb{d}x}}}} \\ {= {\sum\limits_{k = 1}^{L}{\int_{{supp}\quad L_{e,k}}{{L_{e}\left( {x,y} \right)}{f_{r}\left( {x,y,z} \right)}{G\left( {x,y} \right)}{\mathbb{d}x}}}}} \\ {\approx {\underset{k = 1}{\overset{L}{\sum\quad}}{\frac{1}{M_{K}}{\sum\limits_{j = 0}^{M_{k} - 1}{{L_{e}\left( {x_{j},y} \right)}{f_{r}\left( {x_{j},y,z} \right)}{{G\left( {x_{j},y} \right)}.}}}}}} \end{matrix} & (20) \end{matrix}$ where suppL_(e,k) refers to the surface of the respective “k-th” light source. In evaluating the estimate of the integral for the “k-th” light source, for the M_(k)-th query ray, shadow rays determine the fraction of visibility of the area light source, since the point visibility varies much more than the smooth shadow effect. For each light source, the emission L_(e) is attenuated by a geometric term G, which includes the visibility, and the surface properties are given by a bidirectional distribution function f_(r)−f_(s). These integrals are local integrals in the ray tree yielding the value of one node in the ray tree, and can be efficiently evaluated using dependent sampling. In dependent sampling, the query ray comes with the next free integral dimension “d” and the instance “i,” from which the dependent samples are determined in accordance with $\begin{matrix} {x_{j} = {\left( {{{\Phi_{b_{d}}\left( {i,\sigma_{b_{d}}} \right)} \oplus \frac{j}{M_{k}}},{{\Phi_{b_{d + 1}}\left( {i,\sigma_{b_{d + 1}}} \right)} \oplus {\Phi_{2}(j)}}} \right).}} & (21) \end{matrix}$ The offset (Φ_(b) _(d) (i,σ_(b) _(d) ),Φ_(b) _(d+1) (i,σ_(b) _(d+1) )) again is taken from the corresponding generalized scrambled Hammersley point set, which shifts the two-dimensional Hammersley point set $\left( {\frac{j}{M_{k}},{\Phi_{2}(j)}} \right)$ on the light source. Selecting the sample rate M_(k)=2^(n) ^(k) as a power of two, the local minima is obtained for the discrepancy of the Hammersley point set that perfectly stratifies the light source. As an alternative, the light source can be sampled using an arbitrarily-chosen number M_(k) of sample points using $\left( {\frac{j}{M_{k}},{\Phi_{M_{k}}\left( {j,\sigma_{M_{k}}} \right)}} \right)_{j = 0}^{M_{k} - 1}$ as a replication rule. Due to the implicit stratification of the positions of the sample points as described above, the local convergence will be very smooth.

The glossy contribution T_(f) _(e) (L−L_(e)) is determined in a manner similar to that described above in connection with area light sources (equations (20) and (21)), except that a model f_(g) used to simulate glossy scattering will be used instead of the bidirectional distribution function f_(r). In determining the glossy contribution, two-dimensional Hammersley points are generated for a fixed splitting rate M and shifted modulo “one” by the offset (Φ_(b) _(d) (i,σ_(b) _(d) ),Φ_(b) _(d+1) (i,σ_(b) _(d+1) )) taken from the current ray tree depth given by the dimension field “d” of the incoming ray. The ray trees spanned into the scattering direction are decorrelated by assigning the instance fields i′=i+j in a manner similar to that done for simulation of motion blur and depth of field, as described above. The estimates generated for all rays are averaged by the splitting rate “M” and propagated up the ray tree.

Volumetric effects are typically provided by performing a line integration along respective rays from their origins to the nearest surface point in the scene. In providing for a volumetric effect, the computer graphics system 10 generates from the ray data a corresponding offset Φ_(b) _(d) (i) which it then uses to shift the M equidistant samples on the unit interval seen as a one-dimensional torus. In doing so, the rendering time is reduced in comparison to use of an uncorrelated jittering methodology. In addition, such equidistant shifted points typically obtain the best possible discrepancy in one dimension.

Global illumination includes a class of optical effects, such as indirect illumination, diffuse and glossy inter-reflections, caustics and color bleeding, that the computer graphics system 10 simulates in generating an image of objects in a scene. Simulation of global illumination typically involves the evaluation of a rendering equation. For the general form of an illustrative rendering equation useful in global illumination simulation, namely: $\begin{matrix} {{L\left( {\overset{->}{x},\overset{->}{w}} \right)} = {{L_{e}\left( {\overset{->}{x},\overset{->}{w}} \right)} + \quad{\int_{S^{\prime}}{{f\left( {\overset{->}{x},{\overset{->}{w}->\overset{->}{w}}} \right)}{G\left( {\overset{->}{x},{\overset{->}{x}}^{\prime}} \right)}{V\left( {\overset{->}{x},{\overset{->}{x}}^{\prime}} \right)}{L\left( {\overset{->}{x},{\overset{->}{w}}^{\prime}} \right)}{\mathbb{d}A^{\prime}}}}}} & (22) \end{matrix}$ it is recognized that the light radiated at a particular point {right arrow over (x)} in a scene is generally the sum of two components, namely, the amount of light (if any) that is emitted from the point and the amount of light (if any) that originates from all other points and which is reflected or otherwise scattered from the point {right arrow over (x)}. In equation (22), L({right arrow over (x)}, {right arrow over (w)}) represents the radiance at the point {right arrow over (x)} in the direction {right arrow over (w)}=(θ,φ) (where “θ” represents the angle of direction {right arrow over (w)} relative to a direction orthogonal of the surface of the object in the scene containing the point {right arrow over (x)}, and “φ” represents the angle of the component of direction {right arrow over (w)} in a plane tangential to the point {right arrow over (x)}). Similarly, L({right arrow over (x)}′,{right arrow over (w)}′) in the integral represents the radiance at the point {right arrow over (x)}′ in the direction {right arrow over (w)}′=(θ′,φ′) (where “θ′” represents the angle of direction {right arrow over (w)}′ relative to a direction orthogonal of the surface of the object in the scene containing the point {right arrow over (x)}′, and “φ′” represents the angle of the component of direction {right arrow over (w)}′ in a plane tangential to the point {right arrow over (x)}′), and represents the light, if any, that is emitted from point {right arrow over (x)}′ which may be reflected or otherwise scattered from point {right arrow over (x)}.

Continuing with equation (22), L_(e)({right arrow over (x)},{right arrow over (w)}) represents the first component of the sum, namely, the radiance due to emission from the point {right arrow over (x)} in the direction {right arrow over (w)}, and the integral over the sphere S′ represents the second component, namely, the radiance due to scattering of light at point {right arrow over (x)}. ƒ({right arrow over (x)},{right arrow over (w)}′→{right arrow over (w)}) is a bidirectional scattering distribution function which describes how much of the light coming from direction {right arrow over (w)}′ is reflected, refracted or otherwise scattered in the direction {right arrow over (w)}, and is generally the sum of a diffuse component, a glossy component and a specular component. In equation (22), the function G({right arrow over (x)},{right arrow over (x)}′) is a geometric term $\begin{matrix} {{G\left( {\overset{->}{x},{\overset{->}{x}}^{\prime}} \right)} = \frac{\cos\quad\theta\quad\cos\quad\theta^{\prime}}{{{\overset{->}{x} - {\overset{->}{x}}^{\prime}}}^{2}}} & (23) \end{matrix}$ where θ and θ′ are angles relative to the normals of the respective surfaces at points {right arrow over (x)} and {right arrow over (x)}′, respectively. Further in equation (22), V({right arrow over (x)},{right arrow over (x)}′) is a visibility function which equals the value one if the point {right arrow over (x)}′ is visible from the point {right arrow over (x)} and zero if the point {right arrow over (x)}′ is not visible from the point {right arrow over (x)}.

The computer graphics system 10 makes use of global illumination specifically in connection with determination of the diffuse component T_(f) _(d) T_(f) _(d) L and the caustic component T_(f) _(d) T_(f) _(g) _(+f) _(s) L using a photon map technique. Generally, a photon map is constructed by simulating the emission of photons by light source(s) in the scene and tracing the path of each of the photons. For each simulated photon that strikes a surface of an object in the scene, information concerning the simulated photon is stored in a data structure referred to as a photon map, including, for example, the simulated photon's color, position and direction angle. Thereafter a Russian roulette operation is performed to determine the photon's state, that is, whether the simulated photon will be deemed to have been absorbed or reflected by the surface. If a simulated photon is deemed to have been reflected by the surface, the simulated photon's direction is determined using, for example, a bidirectional reflectance distribution function (“BRDF”). If the reflected simulated photon strikes another surface, these operations will be repeated (reference the aforementioned Grabenstein application). The data structure in which information for the various simulated photons is stored may have any convenient form; typically k-dimensional trees, for “k” an integer, are used. After the photon map has been generated, it can be used in rendering the respective components of the image.

In generating a photon map, the computer graphics system 10 simulates photon trajectories, thus avoiding the necessity of discretizing the kernel of the underlying integral equation. The interactions of the photons with the scene, as described above, are stored and used for density estimation. The computer graphics system 10 makes use of a scrambled low-discrepancy strictly-deterministic sequence, such as a scrambled Halton sequence, which has better discrepancy properties in higher dimensions than does an unscrambled sequence. The scrambled sequence also has the benefit, over a random sequence, that the approximation error decreases more smoothly, which will allow for use of an adaptive termination scheme during generation of the estimate of the integral. In addition, since the scrambled sequence is strictly deterministic, generation of estimates can readily be parallelized by assigning certain segments of the low-discrepancy sequence to ones of a plurality of processors, which can operate on portions of the computation independently and in parallel. Since usually the space in which photons will be shot by selecting directions will be much larger than the area of the light sources from which the photons were initially shot, it is advantageous to make use of components of smaller discrepancy, for example, Φ₂ or Φ₃ (where, as above, Φ_(b) refers to the radical inverse function for base “b”), for use in connection with angles at which photons are shot, and components of higher discrepancy, for example, scrambled Φ₅ or Φ₇, for use in connection with sampling of the area of the respective light source, which will facilitate filling the space more uniformly.

The computer graphics system 10 estimates the radiance from the photons in accordance with $\begin{matrix} {{{\overset{\_}{L}}_{r}\left( {x,\omega} \right)} \approx {\frac{1}{A}{\sum\limits_{i \in {B_{k}{(x)}}}{{f_{r}\left( {\omega_{i},x,\omega} \right)}\Phi_{i}}}}} & (24) \end{matrix}$ where, in equation (24), Φ_(i) represents the energy of the respective “i-th” photon, ω_(i) is the direction of incidence of the “i-th photon, “B_(k)(x)” represents the set ofthe “k” nearest photons around the point “x,” and “A” represents an area around point “x” that includes the photons in the set B_(k)(x). Since the energy of a photon is a function of its wavelength, the Φ_(i) in equation (24) also represents the color of the respective “i-th” photon. The computer graphics system 10 makes use of an unbiased but consistent estimator for the area “A” for use in equation (24), which is determined as follows. Given a query ball, that is, a sphere that is centered at point “x” and whose radius r(B_(k)(x)) is the minimal radius necessary for the sphere to include the entire set B_(k)(x), a tangential disk D of radius r(B_(k)(x)) centered on the point “x” is divided into M equal-sized subdomains D_(i), that is $\begin{matrix} {{{{\bigcup_{i = 0}^{M - 1}D_{i}} = {{{D\quad{and}\quad D_{i}}\bigcap D_{j}} \neq {0\quad{for}\quad i} \neq j}},{{{where}{D_{i}}} = {\frac{D}{M} = {\frac{\pi\quad{r^{2}\left( {B_{k}(x)} \right)}}{M}.}}}}\quad} & (25) \end{matrix}$ The set P={D _(i) |D _(i) ∩{x _(i)|_(D) |iεB _(k)(x)}≠0}  (26) contains all the subdomains D_(i) that contain a point x_(i)|_(D) on the disk, which is the position of the “i-th” photon projected onto the plane defined by the disk D along its direction of incidence ω_(i). Preferably, the number M of subdomains will be on the order of {square root}{square root over (k)} and the angular subdivision will be finer than the radial subdivision in order to capture geometry borders. The actual area A is then determined by $\begin{matrix} {A = {\pi\quad{r^{2}\left( {B_{k}(x)} \right)}{\frac{P}{M}.}}} & (27) \end{matrix}$ Determining the actual coverage of the disk D by photons significantly improves the radiance estimate (equation (24)) in corners and on borders, where the area obtained by the standard estimate πr²(B_(k)(x)) would be too small, which would be the case at corners, or too large, which would be the case at borders. In order to avoid blurring of sharp contours of caustics and shadows, the computer graphics system 10 sets the radiance estimate L to black if all domains D_(i) that touch point “x” do not contain any photons.

It will be appreciated that, in regions of high photon density, the “k” nearest photons may lead to a radius r(B_(k)(x) that is nearly zero, which can cause an over-modulation of the estimate. Over-modulation can be avoided by selecting a minimum radius r_(min), which will be used if r(B_(k)(x)) is less than r_(min). In that case, instead of equation (24), the estimate is generated in accordance with $\begin{matrix} {{{\overset{\_}{L}}_{r}\left( {x,\omega} \right)} = {\frac{N}{k}{\sum\limits_{i \in {B_{k}{(x)}}}{\Phi_{i}{f_{r}\left( {\omega_{i},x,\omega_{r}} \right)}}}}} & (28) \end{matrix}$ assuming each photon is started with 1/N of the total flux Φ. The estimator in equation (28) provides an estimate for the mean flux of the “k” photons if r(B_(k)(x))<r_(min).

The global photon map is generally rather coarse and, as a result, subpixel samples can result in identical photon map queries. As a result, the direct visualization of the global photon map is blurry and it is advantageous to perform a smoothing operation in connection therewith In performing such an operation, the computer graphics system 10 performs a local pass integration that removes artifacts of the direct visualization. Accordingly, the computer graphics system 10 generates an approximation for the diffuse illumination term T_(f) _(d) T_(f) _(d) L as $\begin{matrix} \begin{matrix} {{T_{f_{d}}T_{f_{d}}L} \approx {\left( {T_{f_{d}}{\overset{\_}{L}}_{r}} \right)(x)}} \\ {= {\int_{S^{2}{(x)}}{{f_{d}(x)}{{\overset{\_}{L}}_{r}\left( {h\left( {x,\overset{->}{\omega}} \right)} \right)}\cos\quad\theta{\mathbb{d}\omega}}}} \\ {{\approx {\frac{f_{d}(x)}{M}{\sum\limits_{i = 0}^{M - 1}{{\overset{\_}{L}}_{r}\left( {h\left( {x,{\omega\left( {{\arcsin\sqrt{u_{i,1}}},{2\quad\pi\quad u_{i,2}}} \right)}} \right)} \right)}}}},} \end{matrix} & (29) \end{matrix}$ with the integral over the hemisphere S²(x) of incident directions aligned by the surface normal in “x” being evaluated using importance sampling. The computer graphics system 10 stratifies the sample points on a two-dimensional grid by applying dependent trajectory splitting with the Hammersley sequence and thereafter applies irradiance interpolation. Instead of storing the incident flux Φ_(i) of the respective photons, the computer graphics system 10 stores their reflected diffuse power f_(d)(x_(i))Φ_(i) with the respective photons in the photon map, which allows for a more exact approximation than can be obtained by only sampling the diffuse BRDF in the hit points ofthe final gather rays. In addition, the BRDF evaluation is needed only once per photon, saving the evaluations during the final gathering. Instead of sampling the full grid, the computer graphics system 10 uses adaptive sampling, in which refinement is triggered by contrast, distance traveled by the final gather rays in order to more evenly sample the projected solid angle, and the number of photons that are incident form the portion of the projected hemisphere. The computer graphics system fills in positions in the grid that are not sampled by interpolation. The resulting image matrix of the projected hemisphere is median filtered in order to remove weak singularities, after which the approximation is generated. The computer graphics system 10 performs the same operation in connection with, for example, hemispherical sky illumination, spherical high dynamic-range environmental maps, or any other environmental light source.

The computer graphics system 10 processes final gather rays that strike objects that do not cause caustics, such as plane glass windows, by recursive ray tracing. If the hit point of a final gather ray is closer to its origin than a predetermined threshold, the computer graphics system 10 also performs recursive ray tracing. This reduces the likelihood that blurry artifacts will appear in corners, which might otherwise occur since for close hit points the same photons would be collected, which can indirectly expose the blurry structure of the global photon map.

Generally, photon maps have been taken as a snapshot at one point in time, and thus were unsuitable in connection with rendering of motion blur. Following the observation that averaging the result of a plurality of photon maps is generally similar to querying one photon map with the total number of photons from all of the plurality of photon maps, the computer graphics system 10 generates N_(T) photon maps, where N_(T) is determined as described above, at points in time $\begin{matrix} {{t_{b} = {t_{0} + {\frac{b + \frac{1}{2}}{N_{T}}T}}},} & (30) \end{matrix}$ for 0≦b<N_(T). As noted above, N_(T) can equal “one,” in which case “N” photon maps are used, with “N” being chosen as described above. In that case, t _(i) =t ₀+Φ₃(i)T  (31), and thus t_(i,j)=t_(i,0), that is, “t_(i),” for N_(T)=1. In the general case (equation (30)), during rendering, the computer graphics system 10 uses the photon map with the smallest time difference |t_(i,j)−t_(b)| in connection with rendering for the time sample point t_(i,j).

The invention provides a number of advantages. In particular, the invention provides a computer graphics system that makes use of strictly deterministic distributed ray tracing based on low-discrepancy sampling and dependent trajectory splitting in connection with rendering of an image of a scene. Generally, strictly deterministic distributed ray tracing based on deterministic low-discrepancy sampling and dependent trajectory splitting is simpler to implement than an implementation based on random or pseudo-random numbers. Due to the properties of the radical inverse function, stratification of sample points is intrinsic and does not need to be considered independently of the generation of the positions of the sample points. In addition, since the methodology is strictly deterministic, it can be readily parallelized by dividing the image into a plurality of tasks, which can be executed by a plurality of processors in parallel. There is no need to take a step of ensuring that positions of sample points are not correlated, which is generally necessary if a methodology based on random or pseudo-random numbers is to be implemented for processing in parallel.

Moreover, the methodology can be readily implemented in hardware, such as a graphics accelerator, particularly if Hammersley point sets are used, since all points with a fixed index “i” yield a regular grid. A graphics accelerator can render a plurality of partial images corresponding to these regular grids in a number of parallel tasks, and interleave the partial images in an accumulation buffer to provide the final image. Operating in this manner provides very good load balancing among the parallel tasks, since all of the tasks render almost the same image.

In addition, the methodology can readily be extended to facilitate rendering of animations. Generally, an animation consists of a series of frames, each frame comprising an image. In order to decorrelate the various frames, instead of initializing the field of integers used as identifiers for ray instances for each frame by “i,” “i+i_(f)” can be used, where “i_(f)” is a frame number. This operates as an offsetting of “i” by “i_(f),” which is simply a generalization of the Hammersley points. A user can select to initialize the field of integers for each frame by “i,” in which case the frames will not be correlated. In that case, undersampling artifacts caused by smooth motion will remain local and are only smoothly varying. Alternatively, the user can select to initialize the field of integers for each frame by “i+i_(f),” in which case the artifacts will not remain local, and will instead appear as noise or film grain flicker in the final animation. The latter is sometimes a desired feature of the resulting animation, whether for artistic reasons or to match actual film grain. Another variation is to add i_(f) directly to k and clip the result by 2^(n) (reference Code Segment 1, below). In that case, the pixel sampling pattern will change from frame to frame and the frame number i_(f) will need to be known in the post-production process in order to reconstruct the pixel sampling pattern for compositing purposes.

Generally, a computer graphics system that makes use of deterministic low-discrepancy sampling in determination of sample points will perform better than a computer graphics system that makes use of random or pseudo-random sampling, but the performance may degrade to that of a system that makes use of random or pseudo-random sampling in higher dimensions. By providing that the computer graphics system performs dependent splitting by replication, the superior convergence of low-dimensional low-discrepancy sampling can be exploited with the effect that the overall integrand becomes smoother, resulting in better convergence than with stratified random or pseudo-random sampling. Since the computer graphics system also makes use of dependent trajectory sampling by means of infinite low discrepancy sequences, consistent adaptive sampling of, for example, light sources, can also be performed.

In addition, it will be appreciated that, although the computer graphics system has been described as making use of sample points generated using generalized scrambled and/or unscrambled Hammersley and Halton sequences, it will be appreciated that generally any (t,m,s)-net or (t,s)-sequence can be used.

At a more general level, the invention provides an improved quasi-Monte Carlo methodology for evaluating an integral of a function ƒ on the “s” dimensional unit cube [0,1)^(s). In contrast with this methodology, which will be referred to as trajectory splitting by dependent splitting, in prior methodologies, the sample points in the integration domain for which the sample values at which sample values for the function were generated were determined by providing the same number of coordinate samples along each dimension. However, for some dimensions of an integrand, it is often the case that the function ƒ will exhibit a higher variance than for other dimensions. The invention exploits this by making use of trajectory splitting by dependent samples in critical regions.

For example, assume that a function ƒ is to be integrated over s=s₁+s₂ dimensions. The partial integral (equation (32)) $\begin{matrix} {{g(x)} = {{\int_{I^{s_{2}}}{{f\left( {x,y} \right)}{\mathbb{d}y}}} \approx {\frac{1}{N_{2}}{\sum\limits_{j = 0}^{N_{2} - 1}{f\left( {x,y_{j}} \right)}}}}} & (32) \end{matrix}$ (x″ and “y” comprising disjoint sets of the “s” dimensions, and x∪y comprising the set of all of the dimensions), where “N₂” identifies the number of samples selected for the set “y” of dimensions, can be defined over the portion ofthe integration domain that is defined by unit cube [0,1)^(s) ² , which, in turn, corresponds to the portion ofthe integration domain that is associated with set s₂ dimensions. Evaluating g(x) using equation (32) will affect a smoothing ofthe function ƒ in the s₂ dimensions that are associated with set “y.”

The result generated by applying equation (32) can then be used to evaluate the full integral $\begin{matrix} {{{\int_{I^{s_{1}}}{\int_{I^{s_{2}}}{{f\left( {x,y} \right)}{\mathbb{d}y}{\mathbb{d}x}}}} = {{\int_{I^{s_{1}}}{{g(x)}{\mathbb{d}x}}}\quad \approx {\frac{1}{N_{1}}{\underset{i = 0}{\overset{N_{1} - 1}{\sum\quad}}{\frac{1}{N_{2}}{\sum\limits_{j = 0}^{N_{2} - 1}{f\left( {x_{i},y_{j}} \right)}}}}}}},} & (33) \end{matrix}$ where “N₁” identifies the number of samples selected for the set “x” of dimensions, that is, over the remaining dimensions of the integration domain that are associated with the s₁ dimensions that are associated with the set “x.” If the dimension splitting x,y is selected such that the function ƒ exhibits relatively high variance over the set “y” of the integration domain, and relatively low variance over the set “x,” it will not be necessary to generate sample values for the function N₁-times-N₂ times. In that case, it will suffice to generate sample only values N₂ times over the integration domain. If the correlation coefficient of ƒ(ξ,η) and ƒ(ξ,η′), which indicates the degree of correlation between values of function evaluated, for the former, at (x_(i),y_(i))=(ξ,η), and, for the later, at b (x_(i),y_(i))=(ξ,η′), is relatively high, the time comp0lexity required to evaluate the function ƒ_([0,1)) _(s) (x₀, . . . , x_(s−1)) will be decreased.

The smoothness of an integrand can be exploited using a methodology that will be referred to as correlated sampling. Generally, that is, if correlated sampling is not used, in evaluating an integral, each dimension will be associated with its respective sequence. However, in correlated sampling, the same sequence can be used for all of the dimensions over the integration domain, that is $\begin{matrix} {{{\frac{1}{M}{\underset{j - 1}{\overset{N}{\sum\quad}}{\frac{1}{N_{j}}{\sum\limits_{i = 0}^{N_{j} - 1}{f_{j}\left( {x_{i},y_{j}} \right)}}}}} \approx {\frac{1}{M}{\sum\limits_{j = 1}^{M}{\int_{I^{s}}{{f_{j}(x)}{\mathbb{d}x}}}}}}\quad = {{\int_{I^{s}}{\frac{1}{M}{\sum\limits_{j = 1}^{M}{{f_{j}(x)}{\mathbb{d}x}}}}}\quad \approx {\frac{1}{N}{\underset{i = 0}{\overset{N - 1}{\sum\quad}}{\frac{1}{M}{\sum\limits_{j = 1}^{M}{f_{j}\left( x_{i} \right)}}}}}}} & (34) \end{matrix}$

The methodology of trajectory splitting by depending sampling makes use of a combination of the trajectory splitting technique described above in connection with equations (32) and (33) with the correlated sampling methodology described in connection with equation (34).

Since integrals are invariant under toroidal shifting for z_(j)εI^(s) ² , that is, $\begin{matrix} {{\left. \begin{matrix} {{S_{j}:I^{s_{2}}}->I^{s_{2}}} \\ \left. y\mapsto{\left( {y + z_{j}} \right)\quad{mod}\quad 1} \right. \end{matrix}\Rightarrow{\int_{I^{s_{2}}}{{g(y)}{\mathbb{d}y}}} \right. = {\int_{I^{s_{2}}}{{g\left( {S_{j}(y)} \right)}{\mathbb{d}y}}}},} & (35) \end{matrix}$ the values of the integrals also do not change. Thus, if, in equation (33), the inner integral is replicated “M” times, $\begin{matrix} \begin{matrix} {{\int_{I^{s_{1}}}{\int_{I^{s_{2}}}{{f\left( {x,y} \right)}{\mathbb{d}y}{\mathbb{d}x}}}} = {\int_{I^{s_{1}}}{\int_{I^{s_{2}}}{\frac{1}{M}{\sum\limits_{j = 0}^{M - 1}{{f\left( {x,{S_{j}(y)}} \right)}{\mathbb{d}y}{\mathbb{d}x}}}}}}} \\ {\approx {\frac{1}{N}{\underset{i = 0}{\overset{N - 1}{\sum\quad}}{\frac{1}{M}{\sum\limits_{j = 0}^{M - 1}{f\left( {x_{i},{S_{j}\left( y_{i} \right)}} \right)}}}}}} \\ {= {\frac{1}{N}\underset{i = 0}{\overset{N - 1}{\sum\quad}}\frac{1}{M}{\sum\limits_{j = 0}^{M - 1}{{f\left( {x_{i},{\left( {y_{i} + z_{j}} \right){mod}\quad 1}} \right)}.}}}} \end{matrix} & (36) \end{matrix}$ For index “j,” the functions ƒ(x_(i),S_(j)(y_(i))) are correlated, enabling the smoothness of the integrand in those dimensions that are represented by “y” to be exploited, as illustrated above in connection with equation (19) (lens sampling), equations (20) and (21) (area light sources) and equation (29) (approximation for the diffuse illumination term). It will be appreciated that the evaluation using the replication is the repeated application of the local quadrature rule U_(M,s) _(s) :=(z_(j))_(j=0) ^(M) shifted by random offset values y_(i). The use of dependent variables in this manner pays off particularly if there is some smoothness in the integrand along one or more dimensions. Splitting can be applied recursively, which yields a history tree, in which each path through the respective history tree represents a trajectory of a particle such as a photon.

The quasi-Monte Carlo methodology of trajectory splitting by dependent sampling makes use of sets of deterministic, low-discrepancy sample points both for the global quadrature rule U_(N,s) ₁ _(+s) ₂ =(x_(i),y_(i))_(i=0) ^(N), that is, integration over all of the dimensions s₁+s₂ comprising the entire integration domain, as well as for the local quadrature rule U_(M,s) _(s) :=(z_(j))_(j=0) ^(M), that is, integration over the dimensions s₂ of the integration domain. The methodology unites splitting and dependent sampling, exploiting the stratification properties of low-discrepancy sampling. Accordingly, it will be possible to concentrate more samples along those dimensions in which the integrand exhibits high levels of variation, and fewer samples along those dimensions in which the integrand exhibits low levels of variation, which reduces the number of sample points at which the function will need to be evaluated. If the methodology is to be applied recursively a plurality of times, it will generally be worthwhile to calculate a series of values z_(j) that are to comprise the set U_(M,s) ₂ . In addition, the methodology may be used along with importance sampling and, if U is an infinite sequence, adaptive sampling. In connection with adaptive sampling, the adaptations will be applied in the replication independently of the sampling rate, so that the algorithm will remain consistent. The low-discrepancy sample points sets U_(N,s) ₁ _(+s) ₂ and U_(M,s) ₂ can be chosen arbitrarily; for example, the sample point set U_(M,s) ₂ can be a projection of sample point set U_(N,s) ₁ _(+s) ₂ . When trajectory splitting is recursively applied to build trajectory trees, generalizing the point set U_(N,s) ₁ _(+s) ₂ for the subsequent branches can be used to decorrelate the separate parts of the respective tree.

Code Segment 1

The following is a code fragment in the C++ programming language for generating the positions of the jittered subpixel sample points x_(i) unsigned short Period, *Sigma; void InitSigma(int n) { unsigned short Inverse, Digit, Bits; Period = 1 << n; Sigma = new unsigned short [Period]; for (unsigned short i = 0; i < Period; i++) { Digit = Period Inverse = 0; for (bits = i; bits; bits >>= 1) { Digit >>= 1; if(Bits & 1) inverse += Digit; }  Sigma[i] = Inverse; } } void SampleSubpixel(unsigned int *i, double *x, double *y, int s_(x), int s_(y)) { int j = s_(x) & (Period − 1); int k = s_(y) & (Period − 1); *i = j * Period + Sigma[k] *x = (double) s_(x) + (double) Sigma[k] / (double) Period; *y = (double) s_(y) + (double) Sigma[j] / (double) Period; }

Code Segment 2

The following is a code fragment in the C++ programming language for generating a ray tree class Ray { int i; //current instance of low discrepancy vector int d; //current integral dimension in ray tree ... } void Shade (Ray& Ray) { Ray next_ray; int i = ray.i; int d = ray.d ... for (int j = 0; j < M; j+ ) { ... // ray set up for recursion $y = \left( {{{\Phi_{b_{d}}\left( {i,\sigma_{b_{d}}} \right)} \oplus \frac{j}{M}},\ldots,{{\Phi_{b_{d + {\Delta d} - 1}}\left( {i,\sigma_{d_{d + {\Delta d} - 1}}} \right)} \oplus {\Phi_{b_{d + {\Delta d} - 2}}\left( {i - \sigma_{b_{d + {\Delta d} - 2}}} \right)}}} \right)$ next_ray = SetUpRay(y); // determine ray parameters by y next_ray.i = I + j; // decorrelation by generalization next_ray.d = d + Δ; //dimension allocation Shade(next_ray); ... } }

It will be appreciated that a system in accordance with the invention can be constructed in whole or in part from special purpose hardware or a general purpose computer system, or any combination thereof, any portion of which may be controlled by a suitable program. Any program may in whole or in part comprise part of or be stored on the system in a conventional manner, or it may in whole or in part be provided in to the system over a network or other mechanism for transferring information in a conventional manner. In addition, it will be appreciated that the system may be operated and/or otherwise controlled by means of information provided by an operator using operator input elements (not shown) which may be connected directly to the system or which may transfer the information to the system over a network or other mechanism for transferring information in a conventional manner.

The foregoing description has been limited to a specific embodiment of this invention. It will be apparent, however, that various variations and modifications may be made to the invention, with the attainment of some or all of the advantages of the invention. It is the object of the appended claims to cover these and such other variations and modifications as come within the true spirit and scope of the invention. 

1. A computer graphics system for generating a pixel value for a pixel in an image, the pixel value being representative of a point in a scene as recorded on an image plane of a simulated camera, the computer graphics system comprising: A. a sample point generator configured to generate a set of sample points, at least one sample points being generated using at least one dependent sample, the at least one dependent sample comprising at least one element of a low-discrepancy sequence offset by at least one element of another low-discrepancy sequence; and B. a function evaluator configured to generate at least one value representing an evaluation of a selected function at one ofthe sample points generated by said sample point generator, the value generated by the function evaluator corresponding to the pixel value.
 2. A computer graphics system as defined in claim 1 in which the sample point generator is configured to generate at least one sample point with the at least one of said low-discrepancy sequences comprising a Hammersley sequence.
 3. A computer graphics system as defined in claim 1 in which the sample point generator is configured to generate the at least sample point with the at least one of said low-discrepancy sequences comprising a Halton sequence.
 4. A computer graphics system as defined in claim 1 in which the sample point generator is configured to generate at least one sample point having two-dimensions, the at least one dependent sample being defined by an element of a two-dimensional Hammersley point set defined as, for sample point “j,” $\left( {\frac{j}{N},{\Phi_{b}(j)}} \right),$ , for selected base “b” and integer “N,” where Φ_(b)(x) is a radical inverse function given by Φ_(b):  N₀− > I ${x = \left. {\sum\limits_{j = 0}^{\infty}{{a_{j}(x)}b^{j}}}\mapsto{\sum\limits_{j = 0}^{\infty}{{a_{j}(x)}b^{{- j} - 1}}} \right.},$ where (a_(j))_(j=0) ^(∞) is the representation of “x” in integer base “b.”
 5. A computer graphics system as defined in claim 4 in which the sample point generator is configured to generate the at least one sample point with the other of said low-discrepancy sequence being defined by (Φ_(b′)(i),Φ_(b″)(i)) for selected integer “i,” where “b′” and “b″” are selected bases, with b′, b″≠b.
 6. A computer graphics system as defined in claim 5 in which the sample point generator is configured to generate the at least one sample point with the other of said low-discrepancy sequence being defined by (Φ_(b′)(i,σ_(b′)),Φ_(b″)(i,σ_(b″))), where σ is a selected integer permutation.
 7. A computer graphics system as defined in claim 6 in which the sample point generator is configured to generate the at least one sample point using a recursive integer permutation starting with permutation σ₂=(0,1) for base b=2, in which a sequence of permutations is defined as follows: (i) for base “b” even, a permutation σ_(b) is generated by first taking values of 2σ_(b/2) and appending values of ${{2\sigma_{\frac{b}{2}}} + 1},$ and (ii) for base “b” odd, a permutation σ_(b) is generated by taking values of σ_(b−1), incrementing each value that is greater than or equal to $\frac{b - 1}{2}$ by one, and inserting value $\frac{b - 1}{2}$ in the middle of the permutation.
 8. A computer graphics system as defined in claim 6 in which the sample point generator is configured to generate the at least one sample point x_(j) in connection with $x_{j} = \left( {{{\Phi_{b^{\prime}}\left( {i,\sigma_{b^{\prime}}} \right)} \oplus \frac{j}{N}},{{\Phi_{b^{''}}\left( {i,\sigma_{b^{''}}} \right)} \oplus {\Phi_{b^{''}}(j)}}} \right)$ where “⊕” refers to addition modulo “one.”
 9. A computer graphics system as defined in claim 5 in which the sample point generator is configured to generate the at least one sample point x_(j) in connection with ${x_{j} = \left( {{{\Phi_{b^{\prime}}(i)} \oplus \frac{j}{N}},{{\Phi_{b^{''}}(i)} \oplus {\Phi_{b^{''}}(j)}}} \right)},$ where “⊕D” refers to addition modulo “one.”
 10. A computer graphics system as defined in claim 1, the computer graphics system being configured to generate the pixel value for an image using a selected ray-tracing methodology in which simulated rays are shot from respective ones of a plurality of subpixels in the pixel, each subpixel having coordinates (s_(x),s_(y)) in an image plane, the sample point generator being configured to map subpixel coordinates (s_(x),s_(y)) onto strata coordinates (j,k):=(s_(x) mod 2^(n), s_(y) mod 2^(n)), from which a ray is to be shot in accordance with ${x_{i} = \left( {{s_{x} + \frac{\sigma(k)}{2^{n}}},{s_{y} + \frac{\sigma(j)}{2^{n}}}} \right)},$ where “i” is an instance number for the ray generated as i=j2^(n)+σ(k) where integer permutation σ(k):=2^(n)Φ_(b)(k) for 0≦k<2^(n) for selected “n,” where Φ_(b)(x) is a radical inverse function given by Φ_(b):  N₀− > I ${x = \left. {\sum\limits_{j = 0}^{\infty}{{a_{j}(x)}b^{j}}}\mapsto{\sum\limits_{j = 0}^{\infty}{{a_{j}(x)}b^{{- j} - 1}}} \right.},$ where (a_(j))_(j=0) ^(∞) is the representation of “x” in a selected integer base “b.”
 11. A computer graphics system as defined in claim 10 in which base “b” is selected to be “two.”
 12. A computer graphics system as defined in claim 1, the computer graphics system being configured to generate the pixel value for an image using a selected ray-tracing methodology in which simulated rays are shot at a sample point representing a respective point in time during a simulated time interval T beginning at simulated time t₀, the sample point generator being configured to generate, for a ray instance “i,” started at a subpixel position x_(i), an offset Φ_(b)(i) into the time interval, where Φ_(b)(x) is a radical inverse function given by Φ_(b):  N₀− > I ${x = \left. {\sum\limits_{j = 0}^{\infty}{{a_{j}(x)}b^{j}}}\mapsto{\sum\limits_{j = 0}^{\infty}{{a_{j}(x)}b^{{- j} - 1}}} \right.},$ where (a_(j))_(j=0) ^(∞) is the representation of “x” in a selected integer base “b.”
 13. A computer graphics system as defined in claim 12 in which base “b” is selected to be “three.”
 14. A computer graphics system as defined in claim 12, the computer graphics system being configured to generate the pixel value for an image using a selected ray-tracing methodology in which simulated rays are shot at a plurality of sample points representing respective points in time during a simulated time interval T beginning at simulated time t₀, the sample point generator being configured to generate, for a ray instance “i,” started at a subpixel position x_(i), in addition to the offset Φ_(b)(i), N_(T)−1 subsequent samples ${\Phi_{b}(i)} + \frac{j}{N_{T}}$ mod 1 are generated for 1<j<N_(T).
 15. A computer graphics system as defined in claim 14 in which base “b” is selected to be “three.”
 16. A computer graphics system as defined in claim 1 in which the simulated camera includes a simulated lens having a selected lens model, the simulated lens having a lens area, the computer graphics system being configured to generate the pixel value for an image using a selected ray-tracing methodology in which simulated rays are shot through at least one selected point on the simulated lens, the sample point generator being configured to generate at least one sample point representing the at least one selected point by mapping a dependent sample ${y_{i,j,k} = \left( {\left( {{\Phi_{b}\left( {i + j} \right)} \oplus \frac{k}{N}} \right),\left( {{\Phi_{b^{\prime}}\left( {i + j} \right)} \oplus {\Phi_{b^{''}}(k)}} \right)} \right)},$ onto the lens area using a suitable transformation, where (Φ_(b)(i+j),Φ_(b′)(i+j)) comprise elements of the one low-discrepancy sequence, and $\left( {\frac{k}{N},{\Phi_{b^{''}}(k)}} \right)$ comprise elements of the other low-discrepancy sequence, where “N,” and “i,” “j” and “k” are selected integers identifying respective ray instances, and where Φ_(b)(x), Φ_(b′)(x) and Φ_(b″)(x) correspond to a radical inverse function given by Φ_(b) : N₀− > I ${x = \left. {\sum\limits_{j = 0}^{\infty}{{a_{j}(x)}b^{j}}}\mapsto{\sum\limits_{j = 0}^{\infty}{{a_{j}(x)}b^{{- j} - 1}}} \right.},$ where (a_(j))_(j=0) ^(∞) is the representation of “x” in a selected integer base “b.”
 17. A computer graphics system as defined in claim 16 in which “b” is selected to be “five,” “b′” is selected to be “seven” and “b″” is selected to be “two.”
 18. A computer graphics system as defined in claim 16 in which the sample point generator is further configured to generate the at least one selected point by mapping a dependent sample ${y_{i,j,k} = \left( {\left( {{\Phi_{b}\left( {{i + j},\sigma_{b}} \right)} \oplus \frac{k}{N}} \right),\left( {{\Phi_{b^{\prime}}\left( {{i + j},\sigma_{b^{\prime}}} \right)} \oplus {\Phi_{b^{''}}(k)}} \right)} \right)},$ onto the lens area using a suitable transformation where, in (Φ_(b)(i+j,σ_(b′)), Φ_(b″)(i+j,σ_(b″))), σ is a selected integer permutation.
 19. A computer graphics system as defined in claim 18 in which the sample point generator is configured to generate the at least one sample point using a recursive integer permutation starting with permutation σ₂=(0,1) for base b=2, in which a sequence of permutations is defined as follows: (i) for base “b” even, a permutation σ_(b) is generated by first taking values of 2σ_(b/2) and appending values of ${{2\sigma_{\frac{b}{2}}} + 1},$ and (ii) for base “b” odd, a permutation σ_(b) is generated by taking values of σ_(b−1), incrementing each value that is greater than or equal to $\frac{b - 1}{2}$ by one, and inserting value $\frac{b - 1}{2}$ in the middle of the permutation.
 20. A computer graphics system as defined in claim 1, the computer graphics system being configured to generate the pixel value for an image using a selected ray-tracing methodology in which simulated rays are shot at a plurality of sample points representing respective points on at least one object in a scene, the sample point generator being configured to generate a sample point x_(j) on said at least one object according to ${x_{j} = \left( {{{\Phi_{b}(i)} \oplus \frac{j}{N}},{{\Phi_{b^{\prime}}(i)} \oplus {\Phi_{b^{''}}(j)}}} \right)},$ where (Φ_(b)(i),Φ_(b′)(i)) comprise elements of the one low-discrepancy sequence, and $\left( {\frac{j}{N},{\Phi_{b^{''}}(j)}} \right)$ comprise elements of the other low-discrepancy sequence, where “N,” and “i” and “j” are selected integers identifying respective ray instances, and where Φ_(b)(x), Φ_(b′)(x) and Φ_(b″)(x) correspond to a radical inverse function given by Φ_(b) : N₀− > I ${x = \left. {\sum\limits_{j = 0}^{\infty}{{a_{j}(x)}b^{j}}}\mapsto{\sum\limits_{j = 0}^{\infty}{{a_{j}(x)}b^{{- j} - 1}}} \right.},$ where (a_(j))_(j=0) ^(∞) is the representation of “x” in a selected integer base “b.”
 21. A computer graphics system as defined in claim 20 in which the sample point generator is configured to generate the at least one sample point with the low-discrepancy sequence being defined by (Φ_(b′)(i,σ_(b′)),Φ_(b″)(i,σ_(b″))), where σ is a selected integer permutation.
 22. A computer graphics system as defined in claim 21 in which the sample point generator is configured to generate the at least one sample point using the recursive integer permutation starting with permutation σ₂=(0,1) for base b=2, in which a sequence of permutations is defined as follows: (i) for base “b” even, a permutation σ_(b) is generated by first taking values of 2σ_(b/2) and appending values of ${{2\sigma_{\frac{b}{2}}} + 1},$ and (ii) for base “b” odd, a permutation σ_(b) is generated by taking values of σ_(b−1), incrementing each value that is greater than or equal to $\frac{b - 1}{2}$ by one, and inserting value $\frac{b - 1}{2}$ in the middle of the permutation.
 23. A computer-implemented method of generating a pixel value for a pixel in an image, the pixel value being representative of a point in a scene as recorded on an image plane of a simulated camera, the method comprising: A. a sample point generation step of generating a set of sample points, at least one sample points being generated using at least one dependent sample, the at least one dependent sample comprising at least one element of a low-discrepancy sequence offset by at least one element of another low-discrepancy sequence; and B. a function evaluation step of generating at least one value representing an evaluation of a selected function at one of the sample points generated during the sample point generation step, the value generated during the function evaluation step corresponding to the pixel value.
 24. A method as defined in claim 23 in which the sample point generation step includes the step of generating at least one sample point with the at least one of said low-discrepancy sequences comprising a Hammersley sequence.
 25. A method as defined in claim 23 in which the sample point generation step includes the step of generating the at least sample point with the at least one of said low-discrepancy sequences comprising a Halton sequence.
 26. A method as defined in claim 23 in which the sample point generation step includes the step of generating at least one sample point having two-dimensions, the at least one dependent sample being defined by an element of a two-dimensional Hammersley point set defined as, for sample point “j,” $\left( {\frac{j}{N},{\Phi_{b}(j)}} \right),$ , for selected base “b” and integer “N,” where Φ_(b)(x) is a radical inverse function given by Φ_(b):N₀− > I ${x = \left. {\sum\limits_{j = 0}^{\infty}{{a_{j}(x)}b^{j}}}\mapsto{\sum\limits_{j = 0}^{\infty}{{a_{j}(x)}b^{{- j} - 1}}} \right.},$ where (a_(j))_(j=0) ^(∞) is the representation of “x” in integer base “b.”
 27. A method as defined in claim 26 in which the sample point generation step includes the step of generating the at least one sample point with the other of said low-discrepancy sequence being defined by (Φ_(b′)(i),Φ_(b″)(i)) for selected integer “i,” where “b′” and “b″” are selected bases, with b′, b″≠b.
 28. A method as defined in claim 27 in which the sample point generation step includes the step of generating the at least one sample point with the other of said low-discrepancy sequence being defined by be (Φ_(b′)(i,σ_(b′)),Φ_(b″)(i,σ_(b″))), where σ is a selected integer permutation.
 29. A method as defined in claim 28 in which the sample point generation step includes the step of generating the integer permutation recursively starting with permutation σ₂=(0,1) for base b=2, in which a sequence of permutations is defined as follows: (i) for base “b” even, a permutation σ_(b) is generated by first taking values of 2σ_(b/2) and appending values of ${{2\sigma_{\frac{b}{2}}} + 1},$ and (ii) for base “b” odd, a permutation σ_(b) is generated by taking values of σ_(b−1), incrementing each value that is greater than or equal to $\frac{b - 1}{2}$ by one, and inserting value $\frac{b - 1}{2}$ in the middle of the permutation.
 30. A method as defined in claim 28 in which the sample point generation step includes the step of generating the at least one sample point x_(j) in connection with ${x_{j} = \left( {{{\Phi_{b^{\prime}}\left( {i,\sigma_{b^{\prime}}} \right)} \oplus \frac{j}{N}},{{\Phi_{b^{''}}\left( {i,\sigma_{b^{''}}} \right)} \oplus {\Phi_{b^{''}}(j)}}} \right)},$ where “⊕” refers to addition modulo “one.”
 31. A method as defined in claim 27 in which the sample point generation step includes the step of generating the at least one sample point x_(j) in connection with ${x_{j} = \left( {{{\Phi_{b^{\prime}}(i)} \oplus \frac{j}{N}},{{\Phi_{b^{''}}(i)} \oplus {\Phi_{b^{''}}(j)}}} \right)},$ where “⊕” refers to addition modulo “one.”
 32. A method as defined in claim 23, the method being configured to generate the pixel value for an image using a selected ray-tracing methodology in which simulated rays are shot from respective ones of a plurality of subpixels in the pixel, each subpixel having coordinates (s_(x,s) _(y)) in an image plane, the sample point generation step including the step of mapping subpixel coordinates (s_(x),s_(y)) onto strata coordinates (j,k):=(s_(x) mod 2^(n), s_(y) mod 2^(n)), from which a ray is to be shot in accordance with ${x_{i} = \left( {{s_{x} + \frac{\sigma(k)}{2^{n}}},{s_{y} + \frac{\sigma(j)}{2^{n}}}} \right)},$ where “i” is an instance number for the ray generated as i=j2^(n)+σ(k) where integer permutation σ(k):=2^(n)Φ_(b)(k) for 0≦k<2^(n) for selected “n,” where Φ_(b)(x) is a radical inverse function given by Φ_(b) : N₀− > I ${x = \left. {\sum\limits_{j = 0}^{\infty}{{a_{j}(x)}b^{j}}}\mapsto{\sum\limits_{j = 0}^{\infty}{{a_{j}(x)}b^{{- j} - 1}}} \right.},$ where (a_(j))_(j=0) ^(∞) is the representation of “x” in a selected integer base “b.”
 33. A method as defined in claim 32 in which base “b” is selected to be “two.”
 34. A method as defined in claim 33, the method being configured to generate the pixel value for an image using a selected ray-tracing methodology in which simulated rays are shot at a sample point representing a respective point in time during a simulated time interval T beginning at simulated time t₀, the sample point generation step including the step of generating, for a ray instance “i,” started at a subpixel position x_(i), an offset Φ_(b)(i) into the time interval, where Φ_(b)(x) is a radical inverse function given by Φ_(b) : N₀− > I ${x = \left. {\sum\limits_{j = 0}^{\infty}{{a_{j}(x)}b^{j}}}\mapsto{\sum\limits_{j = 0}^{\infty}{{a_{j}(x)}b^{{- j} - 1}}} \right.},$ where (a_(j))_(j=0) ^(∞) is the representation of “x” in a selected integer base “b.”
 35. A method as defined in claim 34 in which base “b” is selected to be “three.”
 36. A method as defined in claim 34, the method being configured to generate the pixel value for an image using a selected ray-tracing methodology in which simulated rays are shot at a plurality of sample points representing respective points in time during a simulated time interval T beginning at simulated time t₀, the sample point generation step including the step of generating, for a ray instance “i,” started at a subpixel position x_(i), in addition to the offset Φ_(b)(i), N_(T)−1 subsequent samples ${\Phi_{b}(i)} + \frac{j}{N_{T}}$ mod 1 are generated for 1<j<N_(T).
 37. A method as defined in claim 36 in which base “b” is selected to be “three.”
 38. A method as defined in claim 23 in which the simulated camera includes a simulated lens having a selected lens model, the simulated lens having a lens area, the method being configured to generate the pixel value for an image using a selected ray-tracing methodology in which simulated rays are shot through at least one selected point on the simulated lens, the sample point generation step including the step of generating at least one sample point representing the at least one selected point by mapping a dependent sample ${y_{i,j,k} = \left( {\left( {{\Phi_{b}\left( {i + j} \right)} \oplus \frac{k}{N}} \right),\left( {{\Phi_{b^{\prime}}\left( {i + j} \right)} \oplus {\Phi_{b^{''}}(k)}} \right)} \right)},$ onto the lens area using a suitable transformation, where (Φ_(b)(i+j),Φ_(b′)(i+j)) comprise elements ofthe one low-discrepancy sequence, and $\left( {\frac{k}{N},{\Phi_{b^{''}}(k)}} \right)$ comprise elements of the other low-discrepancy sequence, where “N,” and “i,” “j” and “k” are selected integers identifying respective ray instances, and where Φ_(b)(x), Φ_(b′)(x) and Φ_(b″)(x) correspond to a radical inverse function given by Φ_(b) : N₀− > I ${x = \left. {\sum\limits_{j = 0}^{\infty}{{a_{j}(x)}b^{j}}}\mapsto{\sum\limits_{j = 0}^{\infty}{{a_{j}(x)}b^{{- j} - 1}}} \right.},$ where (a_(j))_(j=0) ^(∞) is the representation of “x” in a selected integer base “b.”
 39. A method as defined in claim 38 in which “b” is selected to be “five,” “b′” is selected to be “seven” and “b″” is selected to be “two.”
 40. A method as defined in claim 38 in which the sample point generation step includes the step of generating the at least one selected point by mapping a dependent sample ${y_{i,j,k} = \left( {\left( {{\Phi_{b}\left( {i + j} \right)} \oplus \frac{k}{N}} \right),\left( {{\Phi_{b^{\prime}}\left( {i + j} \right)} \oplus {\Phi_{b^{''}}(k)}} \right)} \right)},$ onto the lens area using a suitable transformation where, in (Φ_(b)(i+j,σ_(b′)),Φ_(b″)(i+j,σ_(b″))), σ is a selected integer permutation.
 41. A method as defined in claim 40 in which the sample point generation step includes the step of generating the at least one sample point using a recursive integer permutation starting with permutation σ₂=(0,1) for base b=2, in which a sequence of permutations is defined as follows: (i) for base “b” even, a permutation σ_(b) is generated by first taking values of 2σ_(b/2) and appending values of ${{2\sigma_{\frac{b}{2}}} + 1},$ and (ii) for base “b” odd, a permutation σ_(b) is generated by taking values of σ_(b−1), incrementing each value that is greater than or equal to $\frac{b - 1}{2}$ by one, and inserting value $\frac{b - 1}{2}$ in the middle of the permutation.
 42. A method as defined in claim 23, the method being configured to generate the pixel value for an image using a selected ray-tracing methodology in which simulated rays are shot at a plurality of sample points representing respective points on at least one object in a scene, the sample point generation step including the step of generating a sample point x_(j) on said at least one object according to ${x_{j} = \left( {{{\Phi_{b}(i)} \oplus \frac{j}{N}},{{\Phi_{b^{\prime}}(i)} \oplus {\Phi_{b^{''}}(j)}}} \right)},$ where (Φ_(b)(i),Φ_(b′)(i)) comprise elements of the one low-discrepancy sequence, and $\left( {\frac{j}{N},{\Phi_{b^{''}}(j)}} \right)$ comprise elements of the other low-discrepancy sequence, where “N,” and “i” and “j” are selected integers identifying respective ray instances, and where Φ_(b)(x), Φ_(b′)(x) and Φ_(b″)(x) correspond to a radical inverse function given by Φ_(b) : N₀− > I ${x = \left. {\sum\limits_{j = 0}^{\infty}{{a_{j}(x)}b^{j}}}\mapsto{\sum\limits_{j = 0}^{\infty}{{a_{j}(x)}b^{{- j} - 1}}} \right.},$ where (a_(j))_(j=0) ^(∞) is the representation of “x” in a selected integer base “b.”
 43. A method as defined in claim 42 in which the sample point generation step includes the step of generating the at least one sample point with the low-discrepancy sequence being defined by (Φ_(b′)(i,σ_(b′)),Φ_(b″)(i,σ_(b″))), where σ is a selected integer permutation.
 44. A method as defined in claim 43 in which the sample point generation step includes the step of generating the at least one sample point using a recursive integer permutation starting with permutation σ₂=(0,1) for base b=2, in which a sequence of permutations is defined as follows: (i) for base “b” even, a permutation σ_(b) is generated by first taking values of 2σ_(b/2) and appending values of ${{2\sigma_{\frac{b}{2}}} + 1},$ and (ii) for base “b” odd, a permutation σ_(b) is generated by taking values of σ_(b−1), incrementing each value that is greater than or equal to $\frac{b - 1}{2}$ by one, and inserting value $\frac{b - 1}{2}$ in the middle of the permutation.
 45. A computer program product for use in connection with a computer to provide a computer graphics system for generating a pixel value for a pixel in an image, the pixel value being representative of a point in a scene as recorded on an image plane of a simulated camera, the computer program product comprising a computer-readable medium having encoded thereon: A. a sample point generator module configured to enable the computer to generate a set of sample points, at least one sample points being generated using at least one dependent sample, the at least one dependent sample comprising at least one element of a low-discrepancy sequence offset by at least one element of another low-discrepancy sequence; and B. a function evaluator module configured to enable the computer to generate at least one value representing an evaluation of a selected function at one of the sample points generated by said sample point generator, the value generated by the computer corresponding to the pixel value.
 46. A computer program product as defined in claim 45 in which the sample point generator module is configured to enable the computer to generate at least one sample point with the at least one of said low-discrepancy sequences comprising a Hammersley sequence.
 47. A computer program product as defined in claim 45 in which the sample point generator module is configured to enable the computer to generate the at least sample point with the at least one of said low-discrepancy sequences comprising a Haltono sequence.
 48. A computer program product as defined in claim 45 in which the sample point generator module is configured to enable the computer to generate at least one sample point having two-dimensions, the at least one dependent sample being defined by an element of a two-dimensional Hammersley point set defined as, for sample point “j,” $\left( {\frac{j}{N},{\Phi_{b}(j)}} \right),$ for selected base “b” and integer “N,” where Φ_(b)(x) is a radical inverse function given by Φ_(b) : N₀− > I ${x = \left. {\sum\limits_{j = 0}^{\infty}{{a_{j}(x)}b^{j}}}\mapsto{\sum\limits_{j = 0}^{\infty}{{a_{j}(x)}b^{{- j} - 1}}} \right.},$ where (a_(j))_(j=0) ^(∞) is the representation of “x” in integer base “b.”
 49. A computer program product as defined in claim 48 in which the sample point generator module is configured to enable the computer to generate the at least one sample point with the other of said low-discrepancy sequence being defined by (Φ_(b′)(i),Φ_(b″)(i)) for selected integer “i,” where “b′” and “b″” are selected bases, with b′, b″≠b.
 50. A computer program product as defined in claim 49 in which the sample point generator module is configured to enable the computer to generate the at least one sample point with the other of said low-discrepancy sequence being defined by (Φ_(b′)(i,σ_(b′)),Φ_(b″)(i,σ_(b″))), where σ is a selected integer permutation.
 51. A computer program product as defined in claim 50 in which the sample point generator module is configured to enable the computer to generate the at least one sample point using a recursive integer permutation starting with permutation σ₂=(0,1) for base b=2, in which a sequence of permutations is defined as follows: (i) for base “b” even, a permutation σ_(b) is generated by first taking values of 2σ_(b/2) and appending values of ${{2\sigma_{\frac{b}{2}}} + 1},$ and (ii) for base “b” odd, a permutation σ_(b) is generated by taking values of σ_(b−1), incrementing each value that is greater than or equal to $\frac{b - 1}{2}$ by one, and inserting value $\frac{b - 1}{2}$ in the middle of the permutation.
 52. A computer program product as defined in claim 50 in which the sample point generator module is configured to enable the computer to generate the at least one sample point x_(j) in connection with $x_{j} = \left( {{{\Phi_{b^{\prime}}\left( {i,\sigma_{b^{\prime}}} \right)} \oplus \frac{j}{N}},{{\Phi_{b^{''}}\left( {i,\sigma_{b^{''}}} \right)} \oplus {\Phi_{b^{''}}(j)}}} \right)$ where “⊕” refers to addition modulo “one.”
 53. A computer program product as defined in claim 49 in which the sample point generator module is configured to enable the computer to generate the at least one sample point x_(j) in connection with ${x_{j} = \left( {{{\Phi_{b^{\prime}}(i)} \oplus \frac{j}{N}},{{\Phi_{b^{''}}(i)} \oplus {\Phi_{b^{''}}(j)}}} \right)},$ where “⊕” refers to addition modulo “one.”
 54. A computer program product as defined in claim 45, the computer graphics system being configured to generate the pixel value for an image using a selected ray-tracing methodology in which simulated rays are shot from respective ones of a plurality of subpixels in the pixel, each subpixel having coordinates (s_(x),s_(y)) in an image plane, the sample point generator being configured to map subpixel coordinates (s_(x),s_(y)) onto strata coordinates (j,k):=(s_(x) mod 2^(n), s_(y) mod 2^(n)), from which a ray is to be shot in accordance with ${x_{i} = \left( {{s_{x} + \frac{\sigma(k)}{2^{n}}},{s_{y} + \frac{\sigma(j)}{2^{n}}}} \right)},$ where “i” is an instance number for the ray generated as i=j2^(n)+σ(k) where integer permutation σ(k):=2^(n)Φ_(b)(k) for 0≦k<2^(n) for selected “n,” where Φ_(b)(x) is a radical inverse function given by Φ_(b) : N₀− > I ${x = \left. {\sum\limits_{j = 0}^{\infty}{{a_{j}(x)}b^{j}}}\mapsto{\sum\limits_{j = 0}^{\infty}{{a_{j}(x)}b^{{- j} - 1}}} \right.},$ where (a_(j))_(j=0) ^(∞) is the representation of “x” in a selected integer base “b.”
 55. A computer program product as defined in claim 54 in which base “b” is selected to be “two.”
 56. A computer program product as defined in claim 45, the computer graphics system being configured to generate the pixel value for an image using a selected ray-tracing methodology in which simulated rays are shot at a sample point representing a respective point in time during a simulated time interval T beginning at simulated time t₀, the sample point generator module being configured to enable the computer to generate, for a ray instance “i,” started at a subpixel position x_(i), an offset Φ_(b)(i) into the time interval, where Φ_(b)(x) is a radical inverse function given by Φ_(b) : N₀− > I ${x = \left. {\sum\limits_{j = 0}^{\infty}{{a_{j}(x)}b^{j}}}\mapsto{\sum\limits_{j = 0}^{\infty}{{a_{j}(x)}b^{{- j} - 1}}} \right.},$ where (a_(j))_(j=0) ^(∞) is the representation of “x” in a selected integer base “b.”
 57. A computer program product as defined in claim 56 in which base “b” is selected to be “three.”
 58. A computer program product as defined in claim 56, the computer graphics system being configured to generate the pixel value for an image using a selected ray-tracing methodology in which simulated rays are shot at a plurality of sample points representing respective points in time during a simulated time interval T beginning at simulated time t₀, the sample point generator module being configured to enable the computer to generate, for a ray instance “i,” started at a subpixel position x_(i), in addition to the offset Φ_(b)(i), N_(T)−1 subsequent samples ${\Phi_{b}(i)} + \frac{j}{N_{T}}$ mod 1 are generated for 1<j<N_(T).
 59. A computer program product as defined in claim 58 in which base “b” is selected to be “three.”
 60. A computer program product as defined in claim 45 in which the simulated camera includes a simulated lens having a selected lens model, the simulated lens having a lens area, the computer graphics system being configured to generate the pixel value for an image using a selected ray-tracing methodology in which simulated rays are shot through at least one selected point on the simulated lens, the sample point generator module being configured to enable the computer to generate at least one sample point representing the at least one selected point by mapping a dependent sample ${y_{i,j,k} = \left( {\left( {{\Phi_{b}\left( {i + j} \right)} \oplus \frac{k}{N}} \right),\left( {{\Phi_{b^{\prime}}\left( {i + j} \right)} \oplus {\Phi_{b^{''}}(k)}} \right)} \right)},$ onto the lens area using a suitable transformation, where (Φ_(b)(i+j),Φ_(b′)(i+j)) comprise elements ofthe one low-discrepancy sequence, and $\left( {\frac{k}{N},{\Phi_{b^{''}}(k)}} \right)$ comprise elements of the other low-discrepancy sequence, where “N,” and “i,” “j” and “k” are selected integers identifying respective ray instances, and where Φ_(b)(x), Φ_(b′)(x) and Φ_(b″)(x) correspond to a radical inverse function given by Φ_(b):N₀− > I ${x = \left. {\sum\limits_{j = 0}^{\infty}{{a_{j}(x)}b^{j}}}\mapsto{\sum\limits_{j = 0}^{\infty}{{a_{j}(x)}b^{{- j} - 1}}} \right.},$ where (a_(j))_(j=0) ^(∞) is the representation of “x” in a selected integer base “b.”
 61. A computer program product as defined in claim 60 in which “b” is selected to be “five,” “b′” is selected to be “seven” and “b″” is selected to be “two.”
 62. A computer program product as defined in claim 60 in which the sample point generator module is further configured to enable the computer to generate the at least one selected point by mapping a dependent sample ${y_{i,j,k} = \left( {\left( {{\Phi_{b}\left( {{i + j},\sigma_{b}} \right)} \oplus \frac{k}{N}} \right),\left( {{\Phi_{b^{\prime}}\left( {{i + j},\sigma_{b^{\prime}}} \right)} \oplus {\Phi_{b^{''}}(k)}} \right)} \right)},$ onto the lens area using a suitable transformation where, in (Φ_(b)(i+j,σ_(b′)),Φ_(b″)(i+j,σ_(b″))), σ is a selected integer permutation.
 63. A computer program product as defined in claim 62 in which the sample point generator module is configured to enable the computer to generate the at least one sample point using a recursive integer permutation starting with permutation σ₂=(0,1) for base b=2, in which a sequence of permutations is defined as follows: (i) for base “b” even, a permutation σ_(b) is generated by first taking values of 2σ_(b/2) and appending values of ${{2\sigma_{\frac{b}{2}}} + 1},$ and (ii) for base “b” odd, a permutation σ_(b) is generated by taking values of σ_(b−1), incrementing each value that is greater than or equal to $\frac{b - 1}{2}$ by one, and inserting value $\frac{b - 1}{2}$ in the middle of the permutation.
 64. A computer program product as defined in claim 45, the computer graphics system being configured to generate the pixel value for an image using a selected ray-tracing methodology in which simulated rays are shot at a plurality of sample points representing respective points on at least one object in a scene, the sample point generator module being configured to enable the computer to generate a sample point x_(j) on said at least one object according to ${x_{j} = \left( {{{\Phi_{b}(i)} \oplus \frac{j}{N}},{{\Phi_{b^{\prime}}(i)} \oplus {\Phi_{b^{''}}(j)}}} \right)},$ where (Φ_(b)(i),Φ_(b′)(i)) comprise elements of the one low-discrepancy sequence, and $\left( {\frac{j}{N},{\Phi_{b^{''}}(j)}} \right)$ comprise elements ofthe other low-discrepancy sequence, where “N,” and “i” and “j” are selected integers identifying respective ray instances, and where Φ_(b)(x), Φ_(b′)(x) and Φ_(b″)(x) correspond to a radical inverse function given by Φ_(b):N₀− > I ${x = \left. {\sum\limits_{j = 0}^{\infty}{{a_{j}(x)}b^{j}}}\mapsto{\sum\limits_{j = 0}^{\infty}{{a_{j}(x)}b^{{- j} - 1}}} \right.},$ where (a_(j))_(j=0) ^(∞) is the representation of “x” in a selected integer base “b.”
 65. A computer program product as defined in claim 64 in which the sample point generator module is configured to enable the computer to generate the at least one sample point with the low-discrepancy sequence being defined by (Φ_(b′)(i,σ_(b′)),Φ_(b″)(i,σ_(b″))), where σ is a selected integer permutation.
 66. A computer program product as defined in claim 65 in which the sample point generator module is configured to enable the computer to generate the at least one sample point using the recursive integer permutation starting with permutation σ₂=(0,1) for base b=2, in which a sequence of permutations is defined as follows: (i) for base “b” even, a permutation σ_(b) is generated by first taking values of 2σ_(b/2) and appending values of ${{2\sigma_{\frac{b}{2}}} + 1},$ and (ii) for base “b” odd, a permutation σ_(b) is generated by taking values of σ_(b−1), incrementing each value that is greater than or equal to $\frac{b - 1}{2}$ by one, and inserting value $\frac{b - 1}{2}$ in the middle of the permutation. 